The external analysis in the problem of hidden defects and in the prediction of earthquakes

Authors

  • Babeshko V.A. Kuban State University, Krasnodar, Российская Федерация
  • Evdokimova O.V. Southern Scientific Center, Russian Academy of Science, Rostov-on-Don, Российская Федерация
  • Babeshko O.M. Kuban State University, Krasnodar, Российская Федерация

UDC

539.3

DOI:

https://doi.org/10.31429/vestnik-13-2-19-28

Abstract

In this work the new notion of "external analysis", representing mathematical analysis of the objects constructed with the help of exterior algebra, in particular, exterior forms arising when solving of boundary problems with topological method is carried out, is introduced. External analysis in this case has no connection to algebra, being a mathematical analysis needed to solve a problem. The external analysis the part of the study without which the study and solving of a boundary problem with topological method cannot be carried out. It is not associated with the construction of exterior forms for topological manifolds, which refers to the exterior algebra, but with mathematical analysis of the exterior forms that have arisen in the application of topological methods for boundary problems, and has found numerous applications in many studies. This concept has already come into use in many studies, suggesting that the mathematical analysis of exterior forms and of pseudo differential, integral and functional equations arising from these is applied in the research process. In this work a more precise definition of the external analysis is given and its relation to the exterior algebra is considered. Examples of application of the external analysis in the study of boundary problems in the theory of strength and collapse, as well as seismology, quantum mechanics and mathematical physics are given. In particular, the use of external analysis for the study of the preliminary stage and the possibility of occurrence of a starting earthquake because of harmonic oscillations of lithospheric plates and the basis on which they are located, caused by a vertical harmonic influence, is demonstrated. Firstly, the representation of the boundary problem functional equation for the lithospheric plates is constructed, using the method of exterior algebra. Then the mathematical analysis of the exterior forms built for lithospheric plates is carried out, which is called external analysis. As a result of its application the arising pseudo-differential and functional equations are constructed and studied, which has no relation to the exterior algebra. It is shown that in this case the occurrence of the starting earthquake is possible, and the possibility of its occurrence is even higher than in the static case.

Keywords:

block element, factorization, topology, integral and deferential factorization methods, exterior forms, block structures, boundary problems, pseudodifferential equation, plate, earthquake

Acknowledgement

Отдельные фрагменты работы выполнены в рамках реализации Госзадания на 2016 г. проект (0256-2014-0006), Программы президиума РАН 1-33П, проекты с (0256-2015-0088) по (0256-2015-0093), и при поддержке грантов РФФИ (14-08-00404, 15-01-01379, 15-08-01377, 16-41-230214, 16-41-230218, 16-48-230216).

Author Infos

Vladimir A. Babeshko

академик РАН, д-р физ.-мат. наук, зав. кафедрой математического моделирования Кубанского государственного университета, директор Научно-исследовательского центра прогнозирования и предупреждения геоэкологических и техногенных катастроф Кубанского государственного университета, заведующий лабораторией Южного федерального университета

e-mail: babeshko41@mail.ru

Olga V. Evdokimova

д-р физ.-мат. наук, главный научный сотрудник Южного научного центра РАН

e-mail: evdokimova.olga@mail.ru

Olga M. Babeshko

д-р физ.-мат. наук, главный научный сотрудник Научно-исследовательского центра прогнозирования и предупреждения геоэкологических и техногенных катастроф Кубанского государственного университета

e-mail: babeshko49@mail.ru

References

  1. \item Vinogradov I.M. (Ed.) Matematicheskaya entsiklopediya [Mathematical encyclopedia]. Vol. 1. Vneshnyaya algebra [External algebra]. Moscow, Izd-vo Sovetskaya entsiklopediya Press, 1977, 1151 p. (In Russian) \item Efimov N.V. Vvedenie v teoriyu vneshnikh form [Introduction to the theory of exterior forms]. Moscow, Nauka Press, 1977, 88 p. (In Russian)
  2. Zorich V.A. Matematicheskiy analiz [The mathematical analysis]. Part 2. Moscow, MTsNMO Press, 2002, 788 p. (In Russian)
  3. Shabat B.V. Vvedenie v kompleksnyy analiz [Introduction to complex analysis]. Part 2. Moscow, Nauka Press, 1985, 464 p. (In Russian)
  4. Babeshko V.A., Babeshko O.M. Metod faktorizatsii resheniya nekotorykh kraevykh zadach [Factorization method solution of some boundary value problems]. Doklady akademii nauk [Reports of the Academy of Sciences], 2003, vol. 389. no. 2, pp. 184-188. (In Russian)
  5. Babeshko V.A., Evdokimova O.V., Babeshko O.M. O svyazi faktorizatsionnykh metodov s drugimi podkhodami v teorii differentsial'nykh i integral'nykh uravneniy [About the connection factorization methods with other approaches in the theory of differential and integral equations]. Doklady akademii nauk [Reports of the Academy of Sciences], 2015, vol. 463, no 1, pp. 32-35. (In Russian)
  6. Babeshko V.A., Babeshko O.M. Issledovanie kraevykh zadach dvoynoy faktorizatsiey [The study of boundary value problems by double factorization]. Doklady akademii nauk [Reports of the Academy of Sciences], 2005, vol. 403, no. 1, pp. 20-24. (In Russian)
  7. Babeshko V.A., Babeshko O.M. Integral'nye preobrazovaniya i metod faktorizatsii v kraevykh zadachakh [Integral transforms and factorization method in boundary value problems]. Doklady akademii nauk [Reports of the Academy of Sciences], 2005, vol. 403, no. 6, pp. 26-28. (In Russian)
  8. Evdokimova O.V. Babeshko V.A., Babeshko O.M. Ob integral'nom i differentsial'nom metodakh faktorizatsii [About integral and differential factorization methods]. Doklady akademii nauk [Reports of the Academy of Sciences], 2006, vol. 410, no. 2, pp. 168-172. (In Russian)
  9. Evdokimova O.V., Babeshko O.M., Babeshko V.A. O differentsial'nom metode faktorizatsii v neodnorodnykh zadachakh [Differential method of factorization in non-uniform tasks]. Doklady akademii nauk [Reports of the Academy of Sciences], 2008, vol. 418, no. 3, pp. 321-323. (In Russian)
  10. Babeshko V.A., Evdokimova O.V., Babeshko O.M. O differentsial'nom metode faktorizatsii v zadachakh dlya sploshnykh sred [Differential method of factorization in tasks for continuous media]. Doklady akademii nauk [Reports of the Academy of Sciences], 2008, vol. 421, no. 1, pp. 37-40. (In Russian)
  11. Babeshko V.A., Babeshko O.M., Evdokimova O.V. K teorii blochnogo elementa [To the theory of block element]. Doklady akademii nauk [Reports of the Academy of Sciences], 2009, vol. 427, no. 2, pp. 183-186. (In Russian)
  12. Babeshko V.A., Babeshko O.M., Evdokimova O.V. O blochnom elemente v forme proizvol'noy treugol'noy piramidy [On the block element in the form of arbitrary triangular pyramid]. Doklady akademii nauk [Reports of the Academy of Sciences], 2009, vol. 429, no. 6, pp. 758-761. (In Russian)
  13. Babeshko V.A., Evdokimova O.V., Babeshko O.M. The block element method for a block Structure. In Proc. of the 19th European Conference on Fracture. Kazan. 26-31 August, 2012, Transactions. Pp. 53-60.
  14. Babeshko V.A., Evdokimova O.V., Babeshko O.M. Bodies with cracked coatings. In Proc. of the XLI Summer School-Conference "Advanced Problems In Mechanics" St. Petersburg, July 1-6, 2013. Pp. 143-153.
  15. Babeshko V.A., Evdokimova O.V., Babeshko O.M., Kolesnikov M.N., Shestopalov V.L. Bodies with cracked coatings. In Book of abstracts of International Summer School-Conference "Advanced Problems In Mechanics" St. Petersburg, July 1-6, 2013. Pp. 32-33.
  16. Babeshko V.A., Evdokimova O.V., Babeshko O.M. About Coupling of the Block Elements. In Book: Mechanics and Model-Based Control of Advanced Engineering Systems. 2013. Springer, Heidelberg, New York, Dordrecht, London, pp. 31-39.
  17. Babeshko V.A., Evdokimova O.V., Babeshko O.M. Block Element Method in Applications. in Indo-Russian Workshop on Topical Problems in Theoretical and Applied Mechanics. 2013, 11-15 November, Chennai. Book of Articles. ElitePublishing House PVT LTD, pp. 32-39.
  18. Babeshko V.A., Evdokimova O.V., Babeshko O.M. Study of the Covering with Breaks. In Indo-Russian Workshop on Topical Problems in Theoretical and Applied Mechanics. 2013, 11-15 November, Chennai. Topical Problems in Theoretical and Applied Mechanics. 2013 Book of Articles. ElitePublishing House PVT LTD, pp. 148-155.
  19. Babeshko V.A., Evdokimova O.V., Babeshko O.M. O skrytykh defektakh v nanostrukturakh, telakh s pokrytiyami i seysmologii [About the hidden defects in nanostructures, bodies with coatings and seismology]. Doklady akademii nauk [Reports of the Academy of Sciences], 2014, vol. 457, no. 1, pp. 45-49. (In Russian)
  20. Babeshko V.A., Evdokimova O.V., Babeshko O.M. Topologicheskie metody v teorii skrytykh defektov i nekotorye anomalii [Topological methods in the theory of hidden defects and some anomalies]. Doklady akademii nauk [Reports of the Academy of Sciences], 2014, vol. 457, vol. 6, pp. 650-655. (In Russian)
  21. Babeshko V.A., Evdokimova O.V., Babeshko O.M. O raznotipnykh pokrytiyakh s defektami v staticheskikh zadachakh seysmologii i nanomaterialakh [On heterogeneous surfaces with defects in static problems of seismology and nanomaterials]. Doklady akademii nauk [Reports of the Academy of Sciences], 2014, vol. 459, no. 6, pp. 41-45. (In Russian)
  22. Babeshko V.A., Evdokimova O.V., Babeshko O.M. Blochnye elementy i analiticheskie resheniya granichnykh zadach dlya sistem differentsial'nykh uravneniy [Block-level elements and analytical solutions of boundary-value problems for systems of differential equations]. Doklady akademii nauk [Reports of the Academy of Sciences], 2014, vol. 454, no. 2, pp. 163-167. (In Russian)
  23. Babeshko V.A., Evdokimova O.V., Babeshko O.M. Ob odnoy faktorizatsionnoy zadache Gil'berta-Vinera i metode blochnogo elementa [About one factorization the problem of Hilbert-Wiener method block element]. Doklady akademii nauk [Reports of the Academy of Sciences], 2014, vol. 459, no. 5, pp. 557-561.
  24. Babeshko V.A., Evdokimova O.V., Babeshko O.M. O skrytykh defektakh v nanostrukturakh, telakh s pokrytiyami i seysmologii [About the hidden defects in nanostructures, bodies with coatings and seismology]. Doklady akademii nauk [Reports of the Academy of Sciences], 2014, vol. 457, no. 1, pp. 45-49. (In Russian)
  25. Babeshko V.A., Evdokimova O.V., Babeshko O.M. Gorshkova E.M., Zaretskaya A.V., Mukhin A.S., Pavlova A.V. O konvergentnykh svoystvakh blochnykh elementov [About the convergent properties of block elements]. Doklady akademii nauk [Reports of the Academy of Sciences], 2015, vol. 465, no. 3, pp. 298-301. (In Russian)
  26. Babeshko V.A. Evdokimova O.V. Babeshko O.M. Russia Directed antenna in block structure. In Proc. of the XLIII Summer School - Conference "Advanced Problems in Mechanics" June 22-27, 2015, St. Petersburg. Pp. 39-48.
  27. Babeshko V.A., Rittser A. Ob osobennostyakh metoda blochnogo elementa [About the features of the method block element]. Doklady akademii nauk [Reports of the Academy of Sciences], 2011, vol. 437, no. 2, pp. 182-185. (In Russian)
  28. Babeshko V.A., Evdokimova O.V., Babeshko O.M., Ritzer J. Fedorenko A.G. Topological Approach to the Theory of Prognosis of Seismicity Based on a Mechanical Conception. Doklady Physics, 2013, vol. 58, no. 5, pp. 181-185.
  29. Babeshko V.A., Evdokimova O.V., Babeshko O.M. K probleme fiziko-mekhanicheskogo predvestnika startovogo zemletryaseniya: mesto, vremya, intensivnost' [The problem of physico-mechanical harbinger of starting an earthquake: place, time, intensity]. Doklady akademii nauk [Reports of the Academy of Sciences], 2016, vol. 466, no. 6, pp. 664-669. (In Russian)
  30. Babeshko V.A., Evdokimova O.V., Babeshko O.M. The Theory of the Starting Earthquake. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2016, no. 1, vol. 2, pp. 37-80.
  31. Vorovich I.I., Babeshko V.A. Dinamicheskie smeshannye zadachi teorii uprugosti dlya neklassicheskikh oblastey [Dynamic mixed problem of elasticity theory for nonclassical fields]. Moscow, Nauka Press, 1979, 320 p. (In Russian)
  32. Vorovich I.I., Babeshko V.A., Pryakhina O.D. Dinamika massivnykh tel i rezonansnye yavleniya v deformiruemykh sredakh [Dynamics of massive bodies and resonance phenomena in deformable media]. Moscow, Nauka Press, 1999, 246 p.

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Issue

2016. No. 2

Pages

19-28

Submitted

2016-06-27

Published

2016-06-30

How to Cite

Babeshko V.A., Evdokimova O.V., Babeshko O.M. The external analysis in the problem of hidden defects and in the prediction of earthquakes. Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation, 2016, no. 2, pp. 19-28. DOI: https://doi.org/10.31429/vestnik-13-2-19-28 (In Russian)

Copyright (c) 2016 Babeshko V.A., Evdokimova O.V., Babeshko O.M.

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